The Kervaire–Milnor invariant in the stableclassification of spin 4-manifolds

We consider the role of the Kervaire–Milnor invariant in the classification of closed, connected, spin 4-manifolds, typically denoted by M, up to stabilisation by connected sums with copies of S2×S2. This stable classification is detected by a spin bordism group over the classifying space Bπ of the fundamental group. Part of the computation of this bordism group via an Atiyah–Hirzebruch spectral sequence is determined by a collection of codimension-two Arf invariants. We show that these Arf invariants can be computed by the Kervaire–Milnor invariant evaluated on certain elements of π2(M). In particular this yields a new stable classification of spin 4-manifolds with 2-dimensional fundamental groups, namely those for which Bπ admits a finite 2-dimensional CW-complex model.